Monte Carlo evidence for the deviation from the Alexander-Orbach rule in three-dimensional percolation

Computer simulation is used to study the diffusion at the percolation threshold on large simple cubic lattices. The exponentk for the rms displacementr witht inr t_k is found to be smaller than 0.2, while the Alexander-Orbach 4/3 rule for the spectral dimension predictsk=0.201 ± 0.002.on leave from Minnesota Supercomputer Institute and School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455.     

Corrections to scaling for diffusion exponents on three-dimensional percolation systems at criticality

Recent results of Monte Carlo simulations of the ant-in-the-labyrinth method in three-dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz, motivated by inconsistent results that have appeared in the literature. The results are observed to be sensitive to the form of the scaling correction terms. Using a single correction term, we estimate the valuek=0.197±0.004 for the anomalous diffusion exponent at criticality. When two correction terms are included,k=0.200±0.002 is obtained. These new estimates are consistent with known theoretical bounds, with recent series expansion results, and with numerical calculations of the conductance of random resistor networks above criticality.     

Molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions

Random walk simulations based on a molecular trajectory algorithm are performed on critical percolation clusters. The analysis of corrections to scaling is carried out. It has been found that the fractal dimension of the random walk on the incipient infinite cluster is d_w=2.873±0.008 in two dimensions and 3.78 ± 0.02 in three dimensions. If instead the diffusion is averaged over all clusters at the threshold not subject to the infinite restriction, the corresponding critical exponent k is found to be k=0.3307±0.0014 for two-dimensional space and 0.199 ± 0.002 for three-dimensional space. Moreover, in our simulations the asymptotic behaviors of local critical exponents are reached much earlier than in other numerical methods.     

Diffusion of lattice gases without double occupancy on three-dimensional percolation lattices

We examined the diffusion of lattice gases, where double occupancy of sites is excluded, on three-dimensional percolation lattices at the percolation thresholdp _c . The critical exponent for the root-mean-square displacement was determined to bek=0.183±0.010, which is similiar to the result of Roman for the problem of the ant in the labyrinth. Furthermore, we found a plateau value fork at intermediate times for systems with higher concentrations of lattice gas particles.     

Bounds of percolation thresholds in the enhanced binary tree

By studying its subgraphs, it is argued that the lower critical percolation threshold of the enhanced binary tree (EBT) is bounded as p_c1<0.355059, while the upper threshold is bounded both from above and below by 1/2 according to renormalization-group arguments. We also review a correlation analysis in an earlier work, which claimed a significantly higher estimate of p_c2 than 1/2, to show that this analysis in fact gives a consistent result with this bound. Our result confirms that the duality relation between critical thresholds does not hold exactly for the EBT and its dual, possibly due to the lack of transitivity.     

Current distribution on a three-dimensional,bond-diluted,random-resistor network at the percolation threshold

Current and logarithm-current distributions on a three-dimensional random-bond percolation cubic network were studied at the percolation threshold by computer simulations. Predictions of a hierarchical model that combine fractal structure and randomness agree with our numerical simulations. In the thermodynamic limit the logarithm-current distribution exhibits ann(ln(i))i ^1/3 dependence below some characteristic currenti _c. This distribution may scale with lni/lnL, but the data are insufficient to make this a definite conclusion. Due to the small range of lnL considered, a study of the moments does not reveal this behavior and a study of the distribution itself is required.     

多体体系输运理论——反常扩散

主要介绍最近在多体体系输运理论的一些模型和动力学等工作, 特别是一些有关反常扩散方面的工作结果.     

Cluster and percolation inequalities for lattice systems with interactions

For a lattice gas with attractive potentials of finite range we use the inequalities of Fortuin, Kasteleyn, and Ginibre (FKG) to obtain fairly accurate upper and lower bounds on the equilibrium probabilityp(K) of finding the set of sitesK occupied and the adjacent sites unoccupied, i.e., on the probabilities of finding specified clusters. The probability that a given site, say the origin, is empty or belongs to a cluster of at mostl particles is shown to be a nonincreasing function of the fugacityz and the reciprocal temperature=(T) ^–1; hence the percolation probability is a nondecreasing function ofz and. If the forces are not entirely attractive, or if the ensemble is restricted by forbidding clusters larger than a certain size, the FKG inequalities no longer apply, but useful upper and lower bounds onp(K) can still be obtained if the density of the system and the size of the clusterK are not too large. They are obtained from a generalization of the Kirkwood-Salsburg equation, derived by regarding the system as a mixture of different types of cluster, whose only interaction is that they cannot overlap or touch.Research supported in part by AFOSR Grant #2430B.     

A transfer matrix program to calculate the conductivity of random resistor networks

We describe technical details on a new method of calculating the conductivity of random resistor networks which uses transfer matrix ideas. We give a program which calculates the conductivity of three-dimensional bars, and we provide a few comments on the advantages of this method and its performances.     

Probabilistic bond percolation in random arrays

We consider the problem of percolation in a system having sites distributed at random, but in which only a fractionh of the physical overlaps form viable links. We convert this to a site problem on the covering lattice, and then show that in two dimensionsh - 1/S ⁴ forh - 1, andh - 4)S² forh 1, whereS is proportional to the critical percolation radius in the original array. This result reproduces the T^–1/3 behavior for log(conductivity) expected of variable-range hopping and found by numerical methods. It also accounts for the region of transition tor-percolation asT . We make a prediction that in three dimensions,h = 1/8S³ + const/S⁶, but numerical confirmation is lacking for this case. While the argument is not exact, we have demonstrated a novel approach to random systems.Supported by the National Research Council of Canada.     

Agreement percolation and phase coexistence in some Gibbs systems

We extend some relations between percolation and the dependence of Gibbs states on boundary conditions known for Ising ferromagnets to other systems and investigate their general validity: percolation is defined in terms of the agreement of a configuration with one of the ground states of the system. This extension is studied via examples and counterexamples, including the antiferromagnetic Ising and hard-core models on bipartite lattices, Potts models, and many-layered Ising and continuum Widom-Rowlinson models. In particular our results on the hard square lattice model make rigorous observations made by Hu and Mak on the basis of computer simulations. Moreover, we observe that the (naturally defined) clusters of the Widom-Rowlinson model play (for the WR model itself) the same role that the clusters of the Fortuin-Kasteleyn measure play for the ferromagnetic Potts models. The phase transition and percolation in this system can be mapped into the corresponding liquid-vapor transition of a one-component fluid.     

Diffusion in random one-dimensional systems

Diffusion on the one-dimensional lattice is described by a master equation with nearest-neighbor transfer rates (symmetric or asymmetric). The transfer rates associated with bonds are assumed to be independent, equally distributed random variables. Under various conditions on their common distribution the large time behavior of averaged site probabilities and/or related quantities is exhibited.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

Exact Solutions of a Generalized Multi-Fractional Nonlinear Diffusion Equation in Radical Symmetry

This paper is devoted to investigating exact solutions of a generalized fractional nonlinear anomalous diffusion equation in radical symmetry. The presence of external force and absorption is also considered. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solutions, and the solutions found here can have a compact behavior or a long tailed behavior.     

Diffusion dynamics in branched spherical structure

Diffusion on a spherical surface with trapping is a common phenomenon in cell biology and porous systems. In this paper, we study the diffusion dynamics in a branched spherical structure and explore the influence of the geometry of the structure on the diffusion process. The process is a spherical movement that occurs only for a fixed radius and is interspersed with a radial motion inward and outward the sphere. Two scenarios govern the transport process in the spherical cavity: free diffusion and diffusion under external velocity. The diffusion dynamics is described by using the concepts of probability density function (PDF) and mean square displacement (MSD) by Fokker-Planck equation in a spherical coordinate system. The effects of dead ends, sphere curvature, and velocity on PDF and MSD are analyzed numerically in detail. We find a transient non-Gaussian distribution and sub-diffusion regime governing the angular dynamics. The results show that the diffusion dynamics strengthens as the curvature of the spherical surface increases and an external force is exerted in the same direction of the motion.     

Multifractality in elastic percolation

We present numerical results on the distribution of forces in the central-force percolation model at threshold in two dimensions. We conjecture a relation between the multifractal spectrum of scalar and vector percolation that we test for central-foce percolation. This relation is in excellent agreement with our numerical data.     

Theory of percolation in fluids of long molecules

We use the reference interaction site model (RISM) integral equation theory to study the percolation behavior of fluids composed of long molecules. We examine the roles of hard core size and of length-to-width ratio on the percolation threshold. The critical density _c is a nonmonotonic function of these parameters exhibiting competition of different effects. Comparisons with Monte Carlo calculations of others are reasonably good. For critical exponents, the theory yields =2=2 for molecules of any noninfinite lengthL. WhenL is very large, the theory yields _cL ^–2. These predictions compare favorably with observations of the conductivity for random assemblies of conductive fibers. The threshold region where asymptotic scaling holds requires the correlation length (/ _c ) ^–v to be much larger thanL. Evidently, the range of densities in this region diminishes asL increases, requiring that density deviations from _c be no larger thanL ^–2. Otherwise, crossover behavior will be observed.     

Conductivity exponent in three-dimensional percolation by diffusion based on molecular trajectory algorithm and blind-ant rules

Diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method, respectively. The classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods. Our Monte Carlo simulations by the simple lattice random walk method give the conductivity exponent μ/ν=2.32±0.02 for diffusion on the incipient infinite clusters and μ/ν=2.21±0.03 for diffusion on a percolating lattice above the threshold. However, while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold, the result is found to be μ/ν=2.26±0.02. In addition, it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk.     

Infinite clusters in percolation models

The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN ₀, the number of distinct infinite clusters, is either 0, 1, or and the caseN ₀= (which might occur in sufficiently high dimension) is analyzed.Alfred P. Sloan Research Fellow, Research supported in part by National Science Foundation grant No. MCS 77-20683 and by the U.S.-Israel Binational Science Foundation.Research supported in part by the U.S.Israel Binational Science Foundation.